| L(s) = 1 | + 2.64·2-s − 0.999·4-s − 10.5·5-s + 22·7-s − 23.8·8-s − 28.0·10-s − 5.29·11-s + 58.2·14-s − 55.0·16-s − 116.·17-s + 126·19-s + 10.5·20-s − 14.0·22-s − 31.7·23-s − 12.9·25-s − 21.9·28-s − 52.9·29-s + 182·31-s + 44.9·32-s − 308·34-s − 232.·35-s + 86·37-s + 333.·38-s + 252.·40-s − 444.·41-s + 96·43-s + 5.29·44-s + ⋯ |
| L(s) = 1 | + 0.935·2-s − 0.124·4-s − 0.946·5-s + 1.18·7-s − 1.05·8-s − 0.885·10-s − 0.145·11-s + 1.11·14-s − 0.859·16-s − 1.66·17-s + 1.52·19-s + 0.118·20-s − 0.135·22-s − 0.287·23-s − 0.103·25-s − 0.148·28-s − 0.338·29-s + 1.05·31-s + 0.248·32-s − 1.55·34-s − 1.12·35-s + 0.382·37-s + 1.42·38-s + 0.996·40-s − 1.69·41-s + 0.340·43-s + 0.0181·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.271781523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.271781523\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.64T + 8T^{2} \) |
| 5 | \( 1 + 10.5T + 125T^{2} \) |
| 7 | \( 1 - 22T + 343T^{2} \) |
| 11 | \( 1 + 5.29T + 1.33e3T^{2} \) |
| 17 | \( 1 + 116.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 126T + 6.85e3T^{2} \) |
| 23 | \( 1 + 31.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 52.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 182T + 2.97e4T^{2} \) |
| 37 | \( 1 - 86T + 5.06e4T^{2} \) |
| 41 | \( 1 + 444.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 96T + 7.95e4T^{2} \) |
| 47 | \( 1 - 365.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 190.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 587.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 574T + 2.26e5T^{2} \) |
| 67 | \( 1 - 530T + 3.00e5T^{2} \) |
| 71 | \( 1 - 809.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 154T + 3.89e5T^{2} \) |
| 79 | \( 1 + 460T + 4.93e5T^{2} \) |
| 83 | \( 1 + 322.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.43e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 70T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.908992341859088691391376938844, −8.260643465263172470161403501697, −7.53872368836814804198149786343, −6.58808741033774566236028065414, −5.49514254329033978890333910183, −4.78151608601580199751714874148, −4.19961702521650165192086442119, −3.33464338102780769766453889167, −2.15757679539273751285984231839, −0.63046434390970030031513846437,
0.63046434390970030031513846437, 2.15757679539273751285984231839, 3.33464338102780769766453889167, 4.19961702521650165192086442119, 4.78151608601580199751714874148, 5.49514254329033978890333910183, 6.58808741033774566236028065414, 7.53872368836814804198149786343, 8.260643465263172470161403501697, 8.908992341859088691391376938844
